Following the literature, we construct the QAOA and QAMPA gates using \(XY\)  mixers [1]}, [2]} as:
\(\mathcal {U}_{PS}^{nm}(\gamma )=\mathcal {U}_{ZZ}^{nm}(\gamma )&=&e^{i\gamma \left(J_{nm}Z_n Z_m+h_nZ_n\right)}\\\mathcal {U}_{M}^{nm}(\beta )=\mathcal {U}_{XY}^{nm}(\beta )&=&e^{i\beta \left(X_n X_m+Y_n Y_m\right)}\nonumber \\\mathcal {\tilde{U}}_{MP}^{nm}(\gamma ,\beta )&=&e^{i\left[\beta \left(X_n X_m+Y_n Y_m\right)+\gamma (J_{nm}Z_n Z_m+h_nZ_n)\right]}.\nonumber \) 
